What do singers mean when they talk about singing from the diaphragm? They either mean it as a reference for trying to guide you into feeling the right sensations for abdominal breathing, without explaining the underlying processes, or they legitimately don’t know any better.
Besides the very popular diaphragm, the intercostal muscles (the muscles between all of your ribs) also act as breathing muscles. So, although you can’t breathe without your diaphragm, you can use your diaphragm a little less by using the intercostal muscles — that’s usually referred to as chest breathing because it feels like you are mainly using your chest to breathe. Similarly, just using your diaphragm alone is called abdominal breathing.
When the diaphragm contracts, it flattens from a dome shape to a shape resembling a flat muscle disc, creating more space in the lungs, which generates a negative pressure inside the lungs due to which air streams into them — the process of inhalation.
At the same time, however, the space created in the lungs, above the diaphragm, is compensated below: the abdominal organs (liver, kidneys, intestines, etc.) get pushed and squished together; then, the added pressure tries to relieve itself in the front, where your abdominal wall muscles (your “abs”) are holding it all together. The result is that feeling which may be described as “breathing through the diaphragm”, but, of course, no air is actually getting into the abdomen.
You can then contract your abdominal wall muscles to push back on the guts and, thus, indirectly move the still contracted diaphragm, giving you a mechanism for pressure-controlled exhalation — a process which you may have heard of under the misleading name “breath support”.
That’s the whole point of the entire exercise of bringing your breathing down into or through your diaphragm: so you have better mechanical control over the air release during singing, with more control over the airflow, as opposed to in “chest breathing”, where your diaphragm is pushing down less. If the abdominal wall gets less stretched, it’s harder to sufficiently engage the abdominal wall muscles in the same way for air release.
It’s also possible to engage both intercostals and the diaphragm to their fullest in a mixed type of breathing (as in without the detriment of “chest breathing”, where your abs can’t sufficiently “support”), but I assume your teacher’s priority is probably first to sensitize you to those body parts first before getting fancy.
That’s probably your most important lesson in singing: singing terminology as it stands is very nebulous and nonsensical, so you’ve always got to reflect on what each term that describes certain sensations means in a real, physical sense. It doesn’t mean that you have got to give up on working with sensations and imagery, but you need to be aware that these sensations and imagery only work when you’re being guided by someone who actually understands the physical and physiological bases behind each one, which is still a rare thing.